A Probabilistic Characterization of g-Harmonic Functions

TL;DR

This paper defines g-harmonic functions, explores their relation to g-martingales using viscosity solutions, and discusses the nonlinear expectation framework, including a strict converse mean value property result.

Contribution

It introduces a probabilistic characterization of g-harmonic functions and extends the relation to nonlinear expectation settings, including the continuous case.

Findings

Established the relation between g-harmonic functions and g-martingales.

Extended the relation to the nonlinear expectation framework.

Provided a strict converse mean value property result.

Abstract

This paper gives a definition of g-harmonic functions and shows the relation between the g-harmonic functions and g-martingales. It's direct to construct such relation under smooth case, but for continuous case we need the theory of viscosity solution. The results show that under the nonlinear expectation mechanism, we also can get the similar relation between harmonic functions and martingales. Finally, we will give a result about the strict converse problem of mean value property of g-harmonic functions.